Download 1 CHANCE AND MACROEVOLUTION

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CHANCE AND MACROEVOLUTION*
ROBERTA L. MILLSTEIN†‡
Department of Philosophy
California State University, Hayward
___________________________________________________________________________
*Received July 2000; revised August 2000.
†I would like to thank John Beatty, Ron Giere, and Ken Waters for their helpful comments
on earlier versions of this paper. I also thank the anonymous refereees for their constructive
remarks. This work was supported by a Faculty Support Grant from California State
University, Hayward.
‡Send reprint requests to the author, Department of Philosophy, California State University,
Hayward, 25800 Carlos Bee Boulevard, Hayward, CA 94542.
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Abstract
When philosophers of physics explore the nature of chance, they usually look to quantum
mechanics. When philosophers of biology explore the nature of chance, they usually look to
microevolutionary phenomena such as mutation or random drift. What has been largely
overlooked is the role of chance in macroevolution. The stochastic models of paleobiology
employ conceptions of chance that are similar to those at the microevolutionary level, yet
different from the conceptions of chance often associated with quantum mechanics and
Laplacean determinism.
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1. Introduction. Whereas microevolution concerns changes at or below the species level,
macroevolution concerns changes at or above the species level. Paleontologists are often
interested in the macroevolutionary processes that have led to the formation and extinction of
species and higher categories as exhibited in our fossil record (part of the sub-discipline of
paleontology known as paleobiology). In 1972, a group of paleontologists held two
“informal meetings” at the Marine Biological Laboratory in Woods Hole, Massachusetts.
The outgrowth of these and other subsequent meetings was a series of papers suggesting that
stochastic processes might play a large role in the evolution of species and higher orders.
The main tool of the researchers (who came to be known as the “Woods Hole Group”1) was
a computer program known as the MBL program, which used a Monte Carlo approach to
model the macroevolutionary processes of speciation and extinction. Comparing the results
of the model to fossil data, the researchers called into question the ubiquity of so-called
“deterministic” explanations for macroevolutionary phenomena, arguing that stochastic
processes could account for the same data.
I say “so-called” because when paleontologists refer to “deterministic” explanations, they
are referring to a particular kind of explanation rather than to an ontological claim about the
nature of the universe. They do not mean “deterministic” in the Laplacean sense employed
by most philosophers (where a deterministic universe is one in which any given state
uniquely determines a future state). Similarly, proponents of stochastic models do not deny
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Laplacean determinism. Rather, both deterministic and stochastic macroevolutionary
explanations are consistent with either Laplacean determinism or indeterminism. Thus, the
stochasticity of the models does not depend on the stochasticity of the underlying
phenomena; the underlying phenomena could be ontologically random or Laplacean
deterministic (Raup et al. 1973, Gould et al. 1977). The macroevolutionary senses of
“deterministic” and “stochastic” will be discussed further below.
The stochasticity of macroevolutionary processes has gone largely unexamined by
philosophers of biology;2 analyses of the role of chance in evolution focus primarily on
random drift and mutation, quintessentially microevolutionary phenomena. Yet the research
of the Woods Hole group is striking in what it reveals about concept of chance and the nature
of explanation in evolutionary theory. Furthermore, there are many interesting parallels
between chance at the microevolutionary level (i.e., random drift) and chance at the
macroevolutionary level. In this essay, I will explore these issues.
The structure of the essay is as follows. First, I describe the deterministic explanations
that have been traditionally been given in paleobiology, and contrast these explanations with
the stochastic model of the MBL program. Second, I explore the question of what makes the
“stochastic” model “stochastic” by discussing the various conceptions of chance implicit in
the model. Third, I discuss the apparent explanatory pluralism that is suggested by the
addition of stochastic models to the more traditional deterministic explanations. I then
explain how it is possible for the stochastic model to account for phenomena that have been
traditionally explained deterministically, through a discussion of the nature of a Markov
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process. Finally, I summarize the similarities between the random drift and stochastic
macroevolutionary processes, and suggest some broader conclusions about chance within
evolutionary theory.
2. Background: Deterministic Explanations vs. Stochastic Models.
2.1. Deterministic Explanations in Paleobiology. Before we examine the stochastic
macroevolutionary models of the MBL program, it will be helpful to have an understanding
of more traditional, deterministic explanations.3 Peterson (1983) provides an example of this
kind of explanation. Peterson argues that four genera of iguanid lizards (Anolis,
Chamaeleolis, Chamaelinorops, and Phenacosaurus, referred to as anoline lizards or anoles)
have been more successful than other iguanid lizard genera due to their possession of a
subdigital adhesive pad:
The pad complex appears to be a key innovation (sensu Liem, 1973) in the successful
radiation of the group. Compared to animals that must rely on their claws and toe
position for grip, anoles appear to have greater range of usable locomotor substrates.
They can grip or adhere to a perch of almost any diameter (or radius of curvature),
spatial orientation, or surface texture (e.g., surfaces as smooth as glass or as rough as
bark). It may become possible to state the relationship between the pad complex and
physical aspects of the habitat more precisely in the future, but the adhesive pad seems
to be among the morphological adaptations that permit anoles to adopt highly acrobatic
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locomotor behavior and to exploit the spatial heterogeneity of arboreal niches (Peterson
1983, 245).
In this explanation, the phenomenon being explained is the successful radiation of the anoline
lizards, i.e., the greater number of anoline lizard species as compared to other species of
iguanid lizards. (In macroevolutionary contexts, a “radiation” is a marked increase in the
number of sub-groups within a group; in this case, the radiation in question is the increase in
the number of species in the four anoline lizard genera). According to Peterson, there are
more anoline lizard species than other iguanid species because the possession of a subdigital
adhesive pad provided the anoline lizards with a superior grip, giving them greater agility
and enabling them to more fully explore their habitat as compared to lizards which must rely
on claws and toe position. This is referred to as a “deterministic” explanation because it
invokes a specific cause (the presence of the sub-digital adhesive pad) as an explanation for
the macroevolutionary phenomenon in question (the greater radiation of anoline lizards).
Such an explanation is “taxonbounded” in that it does not treat all taxa alike; rather, it claims
that the physical characteristics of some taxa (the anoline lizard genera) caused them to have
a higher rate of branching than other taxa (non-anoline iguanid lizard genera).
Taxonbounded explanations are one kind of deterministic explanation. Another kind of
deterministic explanation is a “timebounded” explanation (Gould et al. 1977). Timebounded
explanations do not treat all times alike. For example, one could give a timebounded
explanation for why mammals were a very small group up until the Tertiary period; prior to
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that time, they were kept small by a successful competitor: the dinosaurs. Once the dinosaurs
became extinct, times were “good” for mammals, and the number of genera expanded (Gould
et al. 1977, 37). Because this explanation presupposes that some times are “better” for a
group then others, it is considered timebounded. Other timebounded explanations claim that
taxa have specifiable life histories – for example, that older taxa are diverse early in their
history, whereas younger taxa in the same group are diverse late in their history (Raup 1985,
42). Since these explanations invoke specific causes (in these cases, specific time periods) as
the explanation for the phenomenon in question, they too are considered deterministic
explanations.4
2.2. The MBL Program: A Stochastic Model for Macroevolution. In contrast to
deterministic explanations, stochastic models of macroevolution are neither taxonbounded
nor timebounded; that is, they “treat all times and taxa alike” (Gould et al. 1977, 25).5 In the
discussion that follows, I will show how the concepts of untimeboundedness and
untaxonboundedness are embodied by the model used in the MBL program. Although
elements of this model have been criticized since it was first proposed, and alternative
models have been proffered, the MBL model (as described in Raup et al. 1973, Gould et al.
1977, and Raup 1977) continues to be cited in textbooks (see, e.g., Futuyma 1986, Ridley
1993) and recent papers (see, e.g., Uhen 1996, Eble 1999). My focus will therefore be on the
model used by the Woods Hole Group in order to: 1) describe the “classic” stochastic model
from which other models are derived, and 2) to delineate the basic elements of stochastic
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models (the untimebounded, untaxonbounded, and Markovian qualities) without getting
bogged down in the details of the differences between different models. With these goals in
mind, in the description of the MBL program that follows, I will discuss the details of the
program only insofar as they pertain to the discussion of stochasticity. For further details of
the program, see Raup et al. 1973 and Gould et al. 1977.
The MBL program attempts to model the evolutionary process by producing simulated
phylogenetic trees. (A phylogenetic tree is a tree that attempts to depict the genealogical
history of a group of organisms). It begins with a single lineage, from which a “life history”
is generated. At each time interval, and for each current lineage, one of three things occurs:
1) The lineage becomes extinct, 2) The lineage splits into two daughter lineages, or 3) The
lineage persists to the next interval without branching.6 Which of these three fates occurs is
determined by a computer-generated random number, based on predetermined probabilities.
There are two main versions of the MBL program. In one version, the “freely floating”
model, the probabilities of branching and extinction are equal (probability of branching =
probability of extinction = 0.1) and unchanging throughout the simulation (Gould et al.
1977). In the other version, the damped-equilibrium model, there is an optimum number of
co-existing lineages for any interval (the same optimum number applies to all intervals), and
the probabilities of branching and extinction are adjusted to obtain this optimum value.7 This
adjustment occurs by raising the probability of branching when the number of lineages is
below the optimum number, and by raising the probability of extinction when the number of
lineages is above the optimum number. Thus, in the initial phase of the program, where the
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number of lineages is few, the probability of branching will exceed the probability of
extinction, causing an “explosion” of lineages (Gould et al. 1977).
Both of these stochastic models can be said to be “untaxonbounded” and
“untimebounded” to some extent. Both are untaxonbounded because the same probabilities
of branching and extinction apply to all current, endpoint lineages in the simulation. Thus,
all taxa are treated alike. This contrasts with the taxonbounded, deterministic explanation
given by Peterson, where the rate of extinction was less for some taxa (the anoline lizards)
than it was for other taxa (non-anoline lizards), and the rate of branching was greater for
some taxa (the anoline lizards) than it was for other taxa (the non-anoline lizards).
The freely floating model is completely untimebounded; the probabilities of branching
and extinction remain the same throughout all the intervals in the simulation.8 The dampedequilibrium model is untimebounded in the sense that the optimum number of lineages
applies to all time intervals. However, it is not completely untimebounded because not all
time intervals are alike; as previously mentioned, in the initial phase of a simulation the
probability of branching exceeds the probability of extinction. However, for the rest of the
simulation, the model is untimebounded; although the probabilities of branching and
extinction change, they do not do so with respect to any particular phase of the simulation.
Thus, all times after the initial phase are treated alike. This contrasts with the timebounded,
deterministic explanation given above, where prior to the Tertiary period, mammals were not
very successful (high extinction rate, low speciation rate), but once dinosaurs became extinct,
times were “better” for mammals (low extinction rate, high speciation rate).
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Thus, explanations that are both untimebounded and untaxonbounded are referred to as
stochastic; explanations that are either timebounded or taxonbounded are referred to as
deterministic. As mentioned above, this sense of determinism is distinct from Laplacean
determinism; deterministic explanations might also be called “determinate cause”
explanations, since they refer to specific causes related to the properties of particular lineages
or time periods.
3. The Stochasticity of Stochastic Models. So, if stochastic models are not a denial of
Laplacean determinism, in what sense can stochastic accounts be said to be stochastic?
Eble’s recent discussion of the various conceptions of chance used in microevolutionary and
macroevolutionary contexts provides a useful starting point. After an analysis of these
conceptions, I will show how they apply to the stochastic macroevolutionary models of the
MBL program.
3.1. Eble’s Conceptions of Chance. Eble (1999) characterizes five (not entirely distinct)
conceptions of chance that are potentially relevant to evolutionary theory. First, chance
sometimes means “uncaused,” implying a denial of Laplacean determinism. This is the
conception frequently associated with quantum mechanics and nonequilibrium
thermodynamics. Second, chance sometimes means “not by design or deliberate plan,” a
conception associated with Paley and natural theology. According to Eble, neither of these
two conceptions is particularly relevant to present-day evolutionary theory.9 Of greater
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relevance, according to Eble, are two conceptions which taken together elaborate what he
calls the “statistical meaning” of chance:
(1) An event occurs at random because it is unpredictable, due to our ignorance of
causes. This ‘ignorance’ interpretation (dating back to Laplace) is probably the most
frequently used. It is usually associated with the assumption of probabilistic behavior
and indiscriminate sampling (viewed as a property of independent events in nature, not
of experimental design…). (2) An event may be said to be random when it is the result
of the confluence of independent causal chains, as in traffic accidents (this
“coincidence” interpretation stems from Aristotle)… In effect, these two versions of
chance imply and complement each other, and both stand as renditions of the statistical
meaning of chance (1999, 76-77).
Eble argues that the fifth conception of chance, the “evolutionary meaning” of chance, is also
highly relevant to present-day evolutionary theory. The evolutionary meaning of chance is
when “events are independent of an organism’s need and of the directionality provided by
natural selection in the process of adaptation” (Eble 1999, 77; italics in original). For
example, the claim that mutations are due to chance usually implies this meaning – chance
mutations are mutations that can be beneficial, harmful, or neutral. They are not the result of
an organism’s need.
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Without taking a position on Eble’s claim that it is primarily the statistical and
evolutionary conceptions of chance that are important in present-day evolutionary theory, I
will follow Eble in focusing my attention on these conceptions. Eble’s conceptual scheme
provides a useful starting point for a discussion of the meaning of chance in evolutionary
theory, but it is in need of further clarification. In particular, there are three conceptions of
chance implicit in Eble’s “statistical meaning,” not two. In the first of the two statistical
meanings of chance in the quote above, Eble conflates “chance as ignorance” with “chance
as indiscriminate sampling.” However, a simple example will show that these two meanings
are not identical. Consider the canonical example of indiscriminate sampling, the
blindfolded sampling of colored balls from an urn. Suppose the balls are identical except for
the fact that some are red and some are green. One person, fully blindfolded, picks balls out
of the urn, while another person watches. We can even suppose that our observer is
Laplace’s demon himself, an observer who is aware of the picker’s mental states and any
other factors that might be relevant to the picking of balls from the urn.10
According to such an observer, the picking of balls would not be chance as ignorance,
since he is fully aware of all the relevant causes. However, would the demon say that he had
observed indiscriminate sampling? In one sense, perhaps not, because the demon would not
claim that each ball was equally likely to be picked – given the mental states of the picker
and the physical distribution of the balls, some balls get picked whereas other balls do not.
However, in another, important sense, the demon would claim that he had observed
indiscriminate sampling – namely, sampling that was indiscriminate with respect to color. In
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other words, whatever else may be causally relevant to which balls were picked, the color of
the balls is not causally relevant to which ball is picked. Thus, although the demon may not
lay claim to indiscriminate sampling simpliciter, he would concede to indiscriminate
sampling with respect to color, in spite of the fact that he is not ignorant of the relevant
causal factors. The demon would acknowledge that the picking of the balls involved “chance
as indiscriminate sampling,” where the sampling occurs with respect to a given property,
although it does not involve “chance as ignorance.”
This example serves to show that “chance as indiscriminate sampling” does not imply
“chance as ignorance.” However, “chance as ignorance” does not imply “chance as
indiscriminate sampling,” either. For example, in discussing the meaning of “chance
variations,” Darwin stated that by “chance” he meant only to acknowledge “our ignorance of
the cause of each particular variation” ([1859] 1964, 131). There was no indiscriminate
sampling involved – nor was there necessarily any “chance as coincidence” – yet Darwin
invokes “chance as ignorance.” I would thus suggest that “chance as ignorance” is a notion
of chance that has applications outside of the statistical realm, and is not properly part of the
statistical meaning of chance.
But what of the two remaining “statistical meanings” of chance – “chance as
indiscriminate sampling” and “chance as coincidence” – do they imply one another?
Consider the example Eble mentions, a traffic accident – “the result of the confluence of
independent causal chains” (1999, 76). In other words, the causal chain of events that leads
one car to be at a particular place and time is independent from the causal chain of events that
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leads a second car to be at the same place and time, leading to the accident. It is hard to see
how this could be construed as indiscriminate sampling, unless one considers a “population”
of cars involved in traffic accidents – “sampling” implies that there is a population in a way
that “coincidence” does not. If a population of cars is examined, one might discover that car
color is causally irrelevant to traffic accidents, so that there would be indiscriminate sampling
with respect to color. On the other hand, one might discover – as American car insurance
companies claim – that red cars are more likely to be involved in traffic accidents than other
car colors. (We can characterize such an event as discriminate sampling, where color is
causally relevant to traffic accidents.) Of course, there may be indiscriminate sampling with
regard to some other property besides color, but there is no guarantee that a population of
coincidental events will involve indiscriminate sampling. Thus, “chance as indiscriminate
sampling” and “chance as coincidence” are not inextricably connected.
What of Eble’s “evolutionary meaning” of chance – not caused by an organism’s need –
how does that relate to the statistical meanings? Eble claims that these meanings are also
distinct, although they are often used simultaneously in evolutionary explanations. In the
discussion that follows, I will argue that Eble (1999) is correct in that these meanings are
sometimes used simultaneously, through an examination of the stochastic macroevolutionary
models of the MBL program. I will also compare and contrast this usage to the conceptions
of chance involved in random drift, an area that Eble (1999) argues uses both the statistical
and evolutionary meanings of chance. However, given the independence of the three
“statistical meanings” I will refer to “chance as indiscriminate sampling,” and “chance as
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coincidence” separately. These meanings may or may not occur in concert with the
Laplacean notion of “chance as ignorance,” and are thus distinct from it.
3.2. Eble’s Chances in Stochastic Macroevolutionary Models. As we saw above, stochastic
models are characterized by explanations that are untimebounded and untaxonbounded.
These characteristics imply that differences between taxa are causally irrelevant to
differences in rates of branching and extinction within the taxa (this is the untaxonbounded
element) and that different time periods are causally irrelevant to differences in rates of
branching and extinction (this is the untimebounded element). In other words, stochastic
models involve “chance as indiscriminate sampling” – sampling that is indiscriminate with
respect to differences in time intervals and taxa. For example, physical differences between
genera do not confer different rates of speciation and extinction upon different species of the
genus (as would be the case in a deterministic, taxonbounded explanation, such as the one
given for anoline lizards above); rather, origination and extinction occur randomly among the
different species of the taxa.
The analogy between random drift and the stochastic macroevolutionary process implicit
in the MBL program is striking; however, they are not identical processes. In random drift,
physical differences between organisms are causally irrelevant to differences in reproductive
success; that is, there is indiscriminate sampling with respect to physical differences between
organisms. An untaxonbounded macroevolutionary explanation, on the other hand, claims
that physical differences between taxa are causally irrelevant to differences in rates of
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branching and extinction within the taxa (differences between the properties of the higher
taxonomic group are not causally relevant to the branching and extinction of the taxa of the
lower category).11 That is, there is indiscriminate sampling with respect to physical
differences between taxa. Both processes involve “chance as indiscriminate sampling,” but
the sampling takes place at different levels; random drift occurs at the microevolutionary
level, whereas the stochastic model we have been discussing pertains to the
macroevolutionary level. Also, there does not seem to be an analog of “untimeboundedness”
in the random drift case. The issue of whether some times are “good” or “bad” for organisms
does not seem to arise in the microevolutionary case as it does in the macroevolutionary.
Nonetheless, stochastic models and random drift share a further similarity: both incorporate
“evolutionary chance.” That is, since neither samples discriminately with respect to physical
differences (as would be the case with natural selection), both processes are directionless
with respect to adaptation. They may lead in an adaptive direction, but they may equally
well lead in a neutral or even harmful direction.
So, stochastic models of evolution are stochastic in the sense that they reflect “chance as
indiscriminate sampling” and “evolutionary chance.” However, they also express “chance as
coincidence.” In denying that macroevolutionary events are determined by specific time
intervals or membership in a taxonomic group, proponents of the MBL program suggest that
evolutionary events such as multiple extinctions are likely to be the result of many “unrelated
or unconnected” causes (Slowinski and Guyer 1989, 908), not a single cause as would be
proposed under a deterministic explanation. For example, Raup and Marshall ask: “Did the
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order Creodonta die out because of the ‘creodontness’ of its genera (meaning small cranial
volume and presumably low intelligence), or did the order die out just because all its genera
went extinct through independent and widely varying causes?” (1980, 10). Thus, the
extinction of each genus of an order might have a single cause, but the causes would be
unrelated (“independent” events), so that there would be no single cause of the extinction of
the order as a whole. If the causes of origination or extinction are independent, then it is
solely by “coincidence,” or chance, that the higher level macroevolutionary event occurs.
The random drift case is somewhat more complicated. Random drift, unlike the
stochastic models, encompasses situations where there seems to be one determinate cause.
For example, a severe drought might decimate a population of artichokes, killing artichokes
without regard to any physical differences between them. This would be random drift
because it is a case where the physical differences between organisms are causally irrelevant
to differences in reproductive success. Because there seems to be one determinate cause, the
drought, as opposed to a stochastic macroevolutionary process where there are many causes,
one might be inclined to suppose that random drift does not involve “chance as coincidence.”
However, that is not the case. If we examine why some artichokes live and some die, we will
probably find (for example) that some happen to be in areas where the soil was richer or
retains more moisture. That these artichokes survive where others do not is a coincidence – a
confluence of independent causal chains. Therefore, even when many causes are not
apparent, random drift invokes “chance as coincidence,” providing for yet another similarity
between it and the stochastic macroevolutionary processes of the MBL program.
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4. The Stochastic Model: An Alternative to Traditional Deterministic Explanations. In
the previous section, we explored the sense in which the processes modeled by the MBL
program can be said to be chance processes. In this section, we will focus on the outcomes of
these processes and the nature of the explanations they provide. Deterministic explanations
have traditionally been provided for seemingly non-random phenomena such as patterns of
taxonomic diversity (for example, extreme increases or decreases in the number of taxa), the
simultaneous increase or decrease of taxonomic diversity in a number of unrelated taxa, and
morphological trends (directional evolutionary changes in physical traits that take place over
a lengthy time period). Proponents of stochastic accounts claim that their models can explain
phenomena that were previously thought to require deterministic explanations (such as the
radiation in the anoline lizard genera discussed above).
4.1. Theoretical Pluralism in Paleobiology. However, it should be noted that while
stochastic models provide an alternative to deterministic explanations, the two kinds of
explanation are not mutually exclusive explanatory paradigms. That is, it may be the case
that some macroevolutionary phenomena are best explained stochastically, whereas other
macroevolutionary phenomena are best explained deterministically. This is similar to the
situation with natural selection and random drift; although natural selection and random drift
can sometimes explain the same phenomena, it is not the case that in accepting a random
drift explanation one must reject all natural selection explanations (and vice versa). Beatty,
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in discussing the debate between neutralists (proponents of random drift) and selectionists
(proponents of natural selection) suggests that there are both specific and general issues at
stake in the debate:
The more specific issues concern whether or not a change in frequency of a specific set
of genetic alternatives can be ascribed to their selective neutrality and the consequent
random drift of their frequencies. Complimentarily, one might expect that the more
general issues concern whether all or no evolutionary changes – or whether all of a
certain kind or none of a certain kind of evolutionary change – are due to random drift
alone. It is important to recognize, however, that even the most general issues
surrounding this version of the importance of random drift do not boil down to
questions of all or none, but to questions of more or less (1984, 199).
That is, the debate between proponents of random drift and proponents of natural selection is
sometimes over whether a specific instance – a specific change in a population, for example
– is due to natural selection or random drift. Alternatively, such debates might discuss
whether the specific evolutionary change was primarily due to random drift or primarily due
to natural selection. At other times, the debate between neutralists and selectionists is over
whether, in general, most evolutionary changes (or, evolutionary changes of a particular
kind) have been due primarily to natural selection, or primarily to random drift.
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With regard to debates between proponents of deterministic explanations and proponents
of stochastic explanations, the situation is strikingly similar.12 Sometimes, disputants
disagree over the proper explanation for a specific phenomenon. For example, Flessa and
Imbrie (1973) and Flessa and Levinton (1977) argue that common diversity patterns across
taxa during the Phanerozoic call for a deterministic explanation, whereas Smith (1977)
maintains that these patterns can be accounted for by a stochastic model. At other times, the
debate concerns the more general issues. As the Woods Hole Group asks:
How different, then, is the real world from the stochastic system? How, in other words,
is the real world “taxonbound” and “timebound” – i.e., in need of specific, causal
explanations involving uniqueness of time and taxon at various stages of earth history?
The answer would seem to be “not very” (Gould et al. 1977, 32).
However, immediately following the claim that the “real world” can be largely explained
stochastically, the Woods Hole Group proceeds to outline three instances in which
deterministic explanations are required. Similarly, one group of deterministic proponents
admits: “There are, of course, stochastic elements in the genesis and development of all
clades, including those that we designate as [the consequence of some condition or event]”
(Stanley et al. 1981, 117). However, these same proponents are careful to circumscribe the
extent of those stochastic elements: “Only within small taxa, such as genera comprising a
small number of species, do chance factors have more than a small probability of prevailing”
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(Stanley et al. 1981, 125). Thus, proponents on both sides assert that their form of
explanation should predominate, while carefully circumscribing a small area where the other
form of explanation is appropriate. Proponents of stochastic models do not assume that all
macroevolutionary phenomena should be explained stochastically; the same is true of
proponents of deterministic explanations. Beatty argues that “disputants [in evolutionary
biology] defend the importance of their favorite modes of evolution without ruling the others
entirely out of the question” (1984, 207); the disputants in this debate do just that. This
suggests that with regard to macroevolutionary explanations, paleontologists are “theoretical
pluralists” in the sense that Beatty elaborates: “A proponent of theoretical pluralism with
respect to a particular domain believes that the domain is essentially heterogeneous, in the
sense that a plurality of theories or mechanisms is required to account for it, different items in
the domain requiring explanations in terms of different theories or mechanisms” (1995, 65;
italics in original).13 They are also engaging in the kind of “relative significance” debate
that Beatty (1995) describes.
4.2. “Non-randomness” Out of Randomness. Thus, it is often the case that proponents of
stochastic models seek not merely to “fill in” where deterministic explanations seem
inappropriate, but rather to replace specific deterministic explanations (without eliminating
deterministic explanations entirely). Yet it seems puzzling that stochastic models would be
able to account for the same sorts of phenomena that deterministic explanations can account
for; intuitively, one would expect very different sorts of results from a stochastic model than
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from a deterministic explanation. For example, you might expect that a stochastic process
would show fluctuations in taxonomic diversity over time, but you would not necessarily
expect it to show a marked increase in the number of taxa. However, the particular kind of
system embodied by the stochastic models – the Markov process – can often result in
seemingly non-random results that we would not intuitively expect from a stochastic process.
In order to understand the nature of a Markov process, we will first examine the simpler case
of random drift.
A Markov process is a process where the conditional probability of a future event given
all the previous events equals the conditional probability of that future event given only the
present event. Thus, in determining the probability of a future event, only the present
outcome is relevant; the particular history that led to the present outcome is irrelevant.
Random drift is a particular kind of Markov process, called a random walk. The easiest way
to understand a random walk is through an ideal coin-tossing game. Let heads represent an
increase of one unit, tails a decrease of one unit, and the probability of heads or tails be
equal. (This probability is known as the transition probability, the probability of changing
from the current state to a subsequent state). We can then describe a sequence of “winnings”
s1, s2,..., sn, (starting at s0=0) where each si is the total amount “won” (positive or negative)
after the ith trial (Feller 1968). This is a Markov process because it does not matter what
sequence of events has led to the present outcome; for example, the sequence heads, heads,
heads, tails is equivalent to the sequence heads, tails, heads, heads; in both cases, our
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winnings have the value 2, and in both cases, our possible values of our winnings after the
next toss are either 1 (if we get tails) or 3 (if we get heads).
Note that in our ideal coin tossing game, for each trial i, si depends on the previous
outcome si -1 and the transition probability, which is 0.5 for an increase of 1 (heads) and 0.5
for a decrease of 1 (tails). So, for example, if our winnings are currently 7 (which would
imply that we had turned up heads 7 more times than we had turned up tails), there are two
possibilities for our next toss of the coin: 6 (if the coin turns up tails) and 8 (if the coin turns
up heads). However, since heads and tails are equally likely, it is just as likely that we will
have winnings of 8 as it is that we will have winnings of 6; there is no tendency to “equalize
out” the number of heads and tails. The outcome of the previous toss thus constrains the
outcomes of future tosses. Each toss is independent, but the outcome of each toss, since it is
added to the outcome of the previous toss, is not independent. Thus it can happen that
outcomes can “walk” in one direction or another; the values of our coin tosses will probably
fluctuate somewhat, but the value may continue to increase overall (although presumably not
indefinitely).
In fact, our winnings may not return to zero as quickly as one might (intuitively) think.
Our intuition suggests that, in a long series of tosses, our winnings would fluctuate from
negative to positive values frequently, so that in any series of tosses, it is likely that our
winnings recently had a value of zero. However, as Feller has demonstrated, in any sequence
of 2n trials, where the 2kth trial is the last trial where the number of heads and tails is equal,
it is just as likely that k = i as k = n - i.14 So, for example, in sequence of 100 coin tosses, it
24
is just as likely that the last point our winnings equalized was at the 98th toss as it is that the
last point our winnings equalized was at the 2nd toss. This somewhat non-intuitive result
implies that “it is quite likely that in a long coin-tossing game one of the players remains
practically the whole time on the winning side, the other on the losing side” (Feller 1968,
81). For example, for a large number of tosses, the probability that our winnings are positive
for 96.7 percent of the time is 0.2; the probability that our winnings are positive for 99.4
percent of the time is 0.1 (Feller 1968, 82).15
Random drift is analogous to the ideal coin tossing game in the following way. Recall
that random drift is a process in which physical differences are causally irrelevant to
differences in reproductive success. So, if random drift occurs in the absence of selection, in
any given generation it is equally likely that one genotype will be as successful as another
genotype (just at it is equally likely that we would get heads or tails). Furthermore, if we
consider each generation as a trial and the number of alleles as the outcome of that trial, the
number of alleles in each generation depends only on the number of alleles of the previous
generation and the various transition probabilities Tij, (where i is the present number of A
alleles, j is the number of A alleles in the next generation, and Tij is the probability of going
from state i to state j). The transition probability is defined by the binomial distribution:
Tij = [(2N)!/(2N - j)!j!] (i/2N)j (1 - i/2N)2N - 1 (Roughgarden 1996, 65-6)
25
To get a general idea of the properties of this distribution, let us look at a transition
probability matrix for a population of size 3 (2N = 6 genes) containing the allele A (Table 1).
Note that for each state i at t, the most probable state at t + 1 is i. (That is, the most likely
result is that the number of alleles will not change from one generation to the next).
However, other transitions occur with a relatively high probability. In particular, note that
when allele A is predominant in the population, it is still fairly likely that it will increase in
frequency in the next generation. For example, in state 4 at time t, the probability of
changing to state 5 at time t + 1 is 0.263, whereas the probability of changing to state 3 at
time t + 1 is 0.219. Similarly, when allele A does not predominate in the population, it is still
relatively likely that it will decrease in the generation. For example, in state 2 at time t, the
probability of going to state 1 at time t + 1 is 0.263, whereas the probability of going to state
3 at time t + 1 is only 0.219. (States 0 and 6 are the cases where no alleles are A’s or all
alleles are A’s, respectively, and thus no change is possible unless there is a mutation).
Thus, as with the ideal coin tossing game, there is no tendency to return to the
equalization point, which for random drift is the point at which all alleles are represented
equally (in a population where there are 2 alleles at a locus, this is the point at which there
are N copies of each allele, or half of the total of 2N alleles). When an allele predominates in
the population, it may continue to increase in the population (or it may decrease). In
addition, since any future generation will be formed from the previous generation, the
frequencies of the present generation will tend to constrain the values of the future
generation. Since frequency values, unlike coin toss values, can change by variable amounts
26
at each time t, future values are not as constrained by the most recent values in the random
drift case as in the coin tossing case. However, the smaller the changes in frequency from
generation to generation, the more future values will be constrained by the value of the
previous generation. Larger frequency changes in a population mean that frequency values
may return towards the equalization point in just one generation, but they also mean that the
frequency may diverge even farther from the equalization point. Thus, as with the ideal coin
tossing game, it is possible for values to “walk” in one direction or the other. Furthermore,
since random drift is a random walk process like the coin tossing game, the chances of one
allele predominating for a long period of time are reasonably high.
In this way, random drift, a Markov process, is able to explain apparent directionality.
Again, the situation for stochastic models of macroevolution is very similar to the
microevolutionary case. Recall that a Markov process is a process in which only the present
state is relevant and past states are irrelevant. The stochastic models we have been
discussing incorporate a kind of Markov process called a branching process (Feller 1968).
The models can be seen as embodying a branching process because at each point in time,
each “particle” (in this case, a taxon) has a certain probability of terminating (going extinct),
branching into two “particles” (producing two new taxa), or continuing into the next time
interval (persistence). Because each future outcome (two new taxa, extinction of taxon, or
persistence of taxon) is dependent on the present taxa without being dependent on any of the
outcomes prior to the present taxa, the branching process of the stochastic model is
Markovian. Similar to the random walk process discussed above, branching processes also
27
have the property that future outcomes are partly dependent on the present taxa and partly
dependent on the transition probability to the future outcome. In the “freely floating” model
described above, the transition probability from the present state to a branching event is 0.1,
the transition probability from the present state to an extinction state is also 0.1, and the
transition probability from the present state to a “persistence state” (i.e., the taxon simply
persists in time without branching or terminating) is 0.8. In the damped-equilibrium model,
transition probabilities change over the course of a simulation. In the beginning of the
simulation, the probability of branching is greater than the probability of termination, but
once the optimum diversity is reached, the probabilities of branching and termination remain
relatively constant and equal.
Despite the fact that the basic model used in the MBL program is a branching process,
the program also uses a random walk model by superimposing a random walk on top of the
branching process. This is done by counting the number of coexisting lineages at each time
interval, where lineages are produced by a branching process. Gould et al. (1977) use this
method in the “freely floating” model described above. As with the random drift case, the
random walk produced by this model differs from the random walk of an ideal coin tossing
game in that the number of coexisting lineages is not limited to incrementing or
decrementing by one unit. For example, if there are four lineages at time t, and at time t + 1
three of them branch while one goes extinct, the number of coexisting lineages increases
from four to six.
28
Above, we saw how, in a random walk, the dependency of a future state on the present
state has a tendency to constrain the future state; values “walk” from the present state to the
future state. Furthermore, there is no general tendency to return to an equalization point; if
the values in the system have been increasing up to the present state, they are no more likely
to decrease at the next state then they are to increase. This is the situation with the “freely
floating” model. For example, if the number of taxa at time t - 2 is 10, the number of taxa at
t - 1 is 12, and the number of taxa at t is 15, the number of taxa at t + 1 will be an increment
or a decrement from the value of 15. The fact that the two previous time intervals
represented an increase in diversity value does not make it any more likely that there will be
a decrease at t + 1. The damped-equilibrium model behaves similarly, except that there is
some tendency to return to the optimum value that the program has set. Note, however, that
since we are dealing with probabilities, even when the diversity exceeds the optimum value,
it may continue to increase in the next time period. It is just less likely to do so (and the
farther away from the optimum value a simulation goes, the less likely such a departure
becomes, by the design of the program). The characteristics of these two models imply that
the values of these simulations may increase (or decrease) for long periods of time without
returning to the equalization point. (Of course, this is less true for the damped-equilibrium
model than it is for the “freely floating” model).
Consequently, when the simulations are run, the patterns of diversity exhibit considerable
variation. Some show what you would intuitively expect from a stochastic model:
fluctuations in diversity, with no marked increases or decreases. Others, however, exhibit
29
patterns where there are marked increases in diversity, or marked decreases in diversity. In
Figure 1, diagrams 2 and 6 are what one might intuitively expect from a stochastic process:
small amounts of changes in diversity over time. Notice, however, the sudden increase in
diversity in diagram 9, similar to that of diagram 12, and the long periods of stability in both
diagram 7 and diagram 19.
We are now in a position to answer the question raised in the first paragraph of this
section, namely, how is it possible that a stochastic model can account for the same
phenomena that were previously thought to require deterministic explanations? The Woods
Hole Group charges that deterministic explanations are sometimes proposed merely because
the phenomena being explained appear to be nonrandom (Gould et al. 1977). However,
stochastic Markov processes can also (with varying degrees of probability, depending on the
situation) produce these same sorts of results. Thus, if seemingly non-random patterns are
the only evidence, stochastic models can often provide explanations of the same phenomena.
As D. C. Fisher notes, “An important lesson of Markovian models is that some degree of
apparent order can arise by chance … As such, directional trends do not necessarily require
explanations framed in terms of single causes acting throughout the duration of the trend”
(1986, 106). It follows that there is an underdetermination of patterns; you cannot determine
whether a stochastic or a deterministic explanation is appropriate just from looking at the
pattern. Because the MBL program models a Markov process, we can expect it to produce a
great variety of patterns; within the history of a lineage, it may undergo periods of increasing
or decreasing diversity. With Markov processes, the production of directionality is not
30
unusual, but rather quite common. Markov processes are processes that may be constrained
by the immediate past; this leads them to produce orderly results regularly.
Thus, the empirical question remains as to how, in any given instance, one is to decide
whether a deterministic or stochastic model provides the better explanation. Stochastic
models have been proposed as explanations for a variety of macroevolutionary phenomena.
There have been studies of phylogenetic patterns of diversity (Raup et al. 1973; Gould et al.
1977; Smith 1977); morphology (Raup and Gould 1974); rates of evolution (Schopf et al.
1975; Schopf 1979; Raup and Marshall 1980; Bookstein 1987); and faunal extinctions
(Schopf 1974, Simberloff 1974). Each of these studies discusses phenomena to which
stochastic models can potentially be applied. My purpose here is simply to show how
stochastic models can provide an alternative to deterministic explanations.
5. Conclusion. There are many parallels between macroevolutionary stochastic models (and
the “opposing” deterministic style of explanation) and random drift (and the “opposing”
process of natural selection). The scare quotes here indicate that in both the
macroevolutionary and the microevolutionary cases, the two forms of explanation are not
really opposing; they may 1) simultaneously explain different aspects of the same
phenomena, 2) be applied to different phenomena by different people, or even 3) be applied
to different phenomena by the same people. Thus, there is theoretical pluralism.
Nonetheless, there is a sense in which the stochastic models have (historically) provided a
31
challenge to more traditional, Darwinian, selectionist explanations, and this aspect cannot be
overlooked.
In addition, the stochastic processes at the microevolutionary level and the
macroevolutionary level can both be represented as a random walk, implying that seemingly
nonrandom patterns can be produced, even though the processes themselves are random.
Given this similarity, it does not seem very farfetched to suggest that the parallels between
the microevolutionary process of random drift and the stochastic macroevolutionary models
are deliberate, although the Woods Hole Group pointed more towards ecology than
population genetics as their inspiration.
The MBL program models stochastic processes that incorporate conceptions of chance
distinct from both “chance as ignorance” and “chance as uncaused”: “chance as
indiscriminate sampling,” “chance as coincidence,” and “evolutionary chance.” These
conceptions of chance play a role at the microevolutionary level as well, in the form of
random drift. Thus, when evolutionary biologists invoke “chance,” it is more than an
admission of ignorance (they may or may not be ignorant of the relevant causes in a
particular case); it is an expression of a particular kind of process. Moreover, this expression
is independent of quantum mechanical conceptions of chance. The examination of chance in
evolution thus provides reason to rethink some of our traditional ideas about chance. It also
raises the intriguing possibility that the evolution of life is stochastic on a grand scale – that
major events in the biological history of our earth are due to chance
32
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Beatty, John (1984), “Chance and Natural Selection”, Philosophy of Science 51: 183-211.
--- (1995), “The Evolutionary Contingency Thesis”, in Gereon Wolters and James Lennox
(eds.), Concepts, Theories, and Rationality in the Biological Sciences. Pittsburgh:
University of Pittsburgh Press, 45-81.
Bookstein, Fred L. (1987), “Random Walk and the Existence of Evolutionary Rates”,
Paleobiology 13: 446-464.
Brandon, Robert and Scott Carson (1996), “The Indeterministic Character of Evolutionary
Theory: No 'No Hidden Variables' Proof But No Room for Determinism Either”,
Philosophy of Science 63: 315-337.
Darwin, Charles ([1859] 1964), On the Origin of Species: A Facsimile of the First Edition.
Cambridge, Massachusetts: Harvard University Press.
Eble, Gunther J. (1999), “On the Dual Nature of Chance in Evolutionary Biology and
Paleobiology”, Paleobiology 25: 75-87.
Feller, William (1968), An Introduction to Probability Theory and Its Applications. New
York: John Wiley & Sons.
Fisher, Daniel C. (1986), “Progress in Organismal Design”, in David M. Raup and David
Jablonski (eds.), Patterns and Processes in the History of Life. Berlin: Springer-Verlag,
99-117.
33
Flessa, Karl W. and John Imbrie (1973), “Evolutionary Pulsations: Evidence from
Phanerozoic Diversity Patterns”, in Donald H. Tarling and S. K. Runcorn (eds.),
Implications of Continental Drift to the Earth Sciences. London: Academic Press, 247285.
Flessa, Karl W. and Jeffrey S. Levinton (1975), “Phanerozoic Diversity Patterns: Tests for
Randomness”, Journal of Geology 83: 239-248.
Futuyma, Douglas J. (1986), Evolutionary Biology. Sunderland: Sinauer Associates.
Gould, Stephen Jay, David M. Raup, J. John Sepkoski, Thomas J. M. Schopf and Daniel S.
Simberloff (1977), “The Shape of Evolution: A Comparison of Real and Random
Clades”, Paleobiology 3: 23-40.
Grantham, Todd A. (1999), “Explanatory Pluralism in Paleobiology”, Philosophy of Science
(PSA 1998 Proceedings) 66 (Supplement): S223-S236.
Peterson, Jane A. (1983), “The Evolution of the Subdigital Pad in Anolis. I. Comparisons
Among the Anoline Genera”, in Anders G. J. Rhodin and Kenneth Miyata (eds.),
Advances in Herpetology and Evolutionary Biology: Essays in Honorof Ernest E.
Williams. Cambridge, MA: Museum of Comparative Zoology, Harvard University, 245283.
Raup, David M. (1977), “Stochastic Models in Evolutionary Paleontology”, in Anthony
Hallam (ed.), Patterns of Evolution As Illustrated by the Fossil Record. Amsterdam:
Elsevier, 59-78.
--- (1985), “Mathematical Models of Cladeogenesis”, Paleobiology 11: 42-52.
34
Raup, David M. and Stephen Jay Gould (1974), “Stochastic Simulation and the Evolution of
Morphology – Towards a Nomothetic Paleontology”, Systematic Zoology 23: 305-322.
Raup, David M., Stephen Jay Gould, Thomas J. M. Schopf and Daniel S. Simberloff (1973),
“Stochastic Models of Phylogeny and the Evolution of Diversity”, Journal of Geology
81: 525-542.
Raup, David M. and Larry G. Marshall (1980), “Variation Between Groups in Evolutionary
Rates: A Statistical Test of Significance”, Paleobiology 6: 9-23.
Ridley, Mark (1993), Evolution. Boston: Blackwell Scientific Publications.
Roughgarden, Jonathan (1996), Theory of Population Genetics and Evolutionary Ecology:
An Introduction. Upper Saddle River, NJ: Prentice Hall.
Schopf, Thomas J. M. (1974), “Permo-Triassic Extinctions: Relation to Sea-Floor
Spreading.”, Journal of Geology 82: 129-143.
--- (1979), “Evolving Paleontological Views on Determinism and Stochastic Approaches”,
Paleobiology 5: 337-352.
Schopf, Thomas J. M., David M. Raup, Stephen Jay Gould and Daniel S. Simberloff (1975),
“Genomic Versus Morphology Rates of Evolution: Influence of Morphologic
Complexity”, Paleobiology 1: 63-70.
Simberloff, Daniel S. (1974), “Permo-Triassic Extinctions: Effects of Area on Biotic
Equilibrium”, Journal of Geology 82: 267-274.
35
Slowinski, Joseph B. and Craig Guyer (1989), “Testing the Stochasticity of Patterns of
Organismal Diversity: An Improved Null Model”, The American Naturalist 134: 907921.
Smith, Charles A. F. (1977), “Diversity Associations as Stochastic Variables”, Paleobiology
3: 41-48.
Stanley, Steven M., Philip W. Signor III, Scott Lidgard and Alan F. Karr (1981), “Natural
Clades Differ From "Random" Clades: Simulations and Analyses”, Paleobiology 7: 115127.
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36
FOOTNOTES
1 The first paper produced by this group was written in 1973 by David Raup, Stephen
Jay Gould, Thomas Schopf, and Daniel Simberloff. Subsequent papers include Raup and
Gould (1974); Schopf, Raup, Gould, and Simberloff (1975); Gould, Raup, J. John Sepkoski,
Jr., Schopf, and Simberloff (1977); Raup (1977); and Schopf (1979).
2 For a notable exception, see Grantham (1999).
3 As I mentioned above, the use of the term “deterministic” here does not imply
Laplacean determinism. Throughout this essay, unless I specify Laplacean determinism, my
usage of the term “deterministic” should be taken to imply the macroevolutionary,
paleontological meaning that I elaborate in this section.
4 It might seem strange to speak of a time period as a cause, particularly in the case of the
dinosaur example given. In speaking in this way, the Woods Hole group seems to be
thinking in terms of abstract models – that is, a model where there are distinct periods of time
where the probability of extinction is different from the probability of branching
(timebounded) versus a model where there are no such distinct periods (untimebounded).
5 It should be noted that I am contrasting a kind of explanation that has commonly
appeared in the literature (deterministic explanation) with a stochastic model that can be used
for evolutionary explanations. In other words, there is no specific “deterministic model,”
although such models have been developed (see Raup 1985 for discussion).
37
6 The model assumes that speciation only occurs when there is branching, a restriction
that would be questioned by those who would argue that speciation can occur through the
accumulation of gradual changes over time (anagenesis).
7 It might seem odd that one of the stochastic models is an equilibrium model. The
reason for the use of an equilibrium model is to see if the successful use of equilibrium
models in ecology and population biology can be replicated in paleontology (Raup et al.
1973). Furthermore, such a model may be more biologically realistic: “The maintenance of
an equilibrium diversity in the present work implies that an adaptive zone or a geographic
area becomes saturated with taxa and remains in a dynamic equilibrium determined by the
opposing forces of branching (speciation) and extinction” (Raup et al. 1973, 529).
8 In the MBL model, the probabilities of branching and extinction are equal as well as
constant. Other stochastic models allow the probabilities of branching and extinction to
differ from one another (Raup 1985).
9 Brandon and Carson (1996) take a different position concerning the quantum
mechanical conception of chance.
10 This supposition suggests that the example is Laplacean deterministic in that the given
state uniquely determines a future state, a position that I will grant for the sake of argument.
11 Note that whereas the “success” of organisms is measured by the number of offspring
they produce, the “success” of a taxon is measured by the number of taxa at the lower level.
38
For example, one genus is more successful than another genus if it contains a greater number
of species (implying that the species of the genus have had either a greater speciation rate, a
lower extinction rate, or both).
12 The situation is similar, but the point of disagreement is not the same. Raup and
Gould, for example, explain that they are “quite consciously avoiding the subject of
evolutionary mechanisms”; they insist that their model is consistent with selection in
randomly fluctuating environments as well as the fixation of mutations by random drift
(1974, 309). Thus, because selection may underlie the stochastic model, the debate between
neutralists and selectionists does not fall along the same lines as the debate between
proponents of deterministic explanations and proponents of stochastic models.
13 Grantham argues for the stronger thesis of “explanatory pluralism” in paleobiology, in
which “both strategies provide correct explanations for the same event” (1999, S225).
14 We use a value of 2k since, of course, our winnings can only be equal when there has
been an even number of tosses.
15 Likewise, the probability that our winnings are negative for 96.7 percent of the time is
0.2; the probability that our winnings are negative for 99.4 percent of the time is 0.1.
39
(a)
3
4
6
2
Time
7
5
11
10
8
9
1
19
18
16
(b)
17
14
12
13
?
15
Time
?
Scale: 20 genera
40
Number of A alleles in generation t + 1
0
1
0
1.000
1
2
3
4
5
6
0.000 0.000 0.000
0.000
0.000 0.000
0.335
0.402 0.201 0.054
0.008
0.001 0.000
2
0.088
0.263 0.329 0.219
0.082
0.016 0.001
Number of A alleles
in generation t
3
0.016
0.094 0.234 0.312
0.234
0.094 0.016
4
0.001
0.016 0.082 0.219
0.329
0.263 0.088
5
0.000
0.001 0.008 0.054
0.201
0.402 0.335
6
0.000
0.000 0.000 0.000
0.000
0.000 1.000
41
FIGURE/TABLE LEGENDS
Figure 1: Comparison of stochastically generated diversity patterns with real diversity
patterns. Numbers above diagrams are for ease of identification only. The width of a
diagram at a given “point” in time represents the number of taxa at that time (more taxa =
greater diversity). (a) Diversity diagrams for one run of the MBL program with probabilities
of branching and extinction set to 0.1. (b) Diversity diagrams for genera within orders of
Brachiopods (From Gould et al. 1977).
Table 1: Transition probability matrix for a population of size N = 3 (2N genes). Numbers
generated by computer program and rounded to the third decimal place.